Methods and systems for adaptive control

ABSTRACT

A system includes a power supply model component and a disturbance estimating component. The power supply model component receives an estimated disturbance. The disturbance estimating component provides said estimated disturbance from a difference between a sensed output of a power supply and an estimated output from said power supply model component.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority of U.S. Provisional Application No.60/735,384 entitled “Adaptive Linear Controller for DC-to-DCConverters,” filed on Nov. 11, 2005, which is incorporated by referenceherein.

BACKGROUND

Applying digital methods to the control of systems bears the promise ofcreating new features, improving performance, providing greater productflexibility, and providing a lower cost. System operatingcharacteristics dictated by a stored program, rather than the parametersof a set of discrete components, can result in cost and space savings aswell as capacity for real time adaptation of those characteristics,greater sophistication in control algorithms and the ability togenerate, store and recall valuable real-time functional data.

However, digital feedback control requires high resolution and highspeed. These requirements have limited the adoption of digital controlin many fields. The advent of low cost logic has it made possible theapplication of digital control techniques to cost sensitive fields. Asthe cost of digital logic decreases, new opportunities arise.

A typical digitally controlled feedback system has an analog to digitalconverter, digital loop compensator, power device driver, and anexternal system to be controlled. An example of a system in whichapplication of digital control can improve performance or lower cost isthe switching power supply or DC-to-DC converter. (However, many othersystems would also benefit from application of digital control.)

It is very desirable to minimize the cost, size and power dissipation ofa low-cost off-line switching power supply for low power applications,such as recharging cells and batteries used in portable consumerappliances, such as entertainment units, personal digital assistants,and cell phones, for example.

A PWM switched power supply requires a variable pulse width that iscontrolled by an error signal derived by comparing actual output voltageto a precise reference voltage. The pulse width of the switchinginterval must also be constrained to be within a minimum and maximumduration. These constraints are imposed for correct PWM power supply ormotor driver operation.

An example of a digitally controlled system is shown in FIG. 1. In theexample shown in FIG. 1, the system is a simple buck Switching powersupply. The fundamental components are the same for any Switching powersupply. The sample system shown in FIG. 1 includes three majorcomponents: a compensator preceded by an ADC, PWM and power switches,and passive LC network.

Most power management design is based on simple, continuous compensationusing frequency domain analysis. Bode analysis is frequently used as thedesign technique of choice.

More modem techniques such as modeling of the converter in discrete timeand using pole placement or optimization techniques to set the gains arenot usually considered. Recently developed digital power managementchips use the digital equivalent of analog continuous time designs. Thedesign procedure starts with an analog prototype which is discretizedand implemented in hardware.

BRIEF SUMMARY

In one embodiment, the system of these teachings includes a Switchingpower supply, an adaptive plant estimation component capable ofreceiving an output voltage from the Switching power supply and an inputto the Switching power supply and of providing a model of the Switchingpower supply; the model reflecting changes in output voltage state ofthe Switching power supply, a compensator design component capable ofreceiving the model of the Switching power supply and of providingcompensator parameters, the compensator parameters reflecting changes inoutput voltage state of the Switching power supply, an adaptivecompensator capable of receiving the compensator parameters and ofproviding the input control signal to the driver component.

In another embodiment, the controller of these teachings includes asampling component capable of sampling an output signal from a systemand an input signals from the system at a first sampling rate, the firstsampling rate being at least equal to a predetermined operating rate, aninput parameter obtaining component capable of receiving the outputsignal and the input signal sampled at the first sampling rate and ofobtaining values for a plurality of input parameters, the values for theinput parameters being sampled at the first sampling rate, a decimatorcomponent capable of receiving the values for the input parameterssampled at the first sampling rate and of providing subsampled valuesfor the input parameters, subsampled values being sampled at a secondsampling rate, the second sampling rate been slower than the firstsampling rate, an adaptive plant estimator component capable ofreceiving the subsampled values of the input parameters and of obtaininga model of the system, the model reflecting variations in the system anda compensator design component capable of receiving the model of thesystem and of providing compensator parameters, the compensatorparameters reflecting changes in the system, values of said compensatorparameters being sampled at the second sampling rate, the compensatordesign component being capable of providing the values of thecompensator parameter to a compensator.

Various other embodiments of the system of the controller of theseteachings are also disclosed. Various embodiments of the method of theseteachings are also disclosed.

For a better understanding of these teachings, together with other andfurther needs thereof, reference is made to the accompanying drawingsand detailed description and its scope will be pointed out in theappended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graphical schematic presentation of a conventional system;

FIG. 2 is a graphical schematic presentation of an embodiment of thesystem of these teachings;

FIG. 3 is a graphical schematic representation of another embodiment ofthe system of these teachings;

FIG. 4 shows a graphical schematic representation of a furtherembodiment of the system of these teachings;

FIG. 5 shows a graphical schematic representation of a component of anembodiment of the system of these teachings;

FIG. 6 depicts a graphical representation of another embodiment of thecomponent of an embodiment of the system of these teachings;

FIG. 7 is a graphical schematic representation of yet another embodimentof a system of these teachings;

FIG. 8 depicts a graphical schematic representation of still a furtherembodiment of the system of these teachings;

FIGS. 9-14 depict graphical representations of results from embodimentsof the system of these teachings;

FIG. 15 is a graphical schematic presentation of a state space estimatorwhere, in the most general formulation, the signals are vectors and thegains are matrices;

FIG. 16 is a graphical representations of results showing the responseof the state estimator for a Switching power supply where the statevariables are capacitor voltage and inductor current;

FIG. 17 a is a graphical schematic representation of another estimatorstructure embodiment of these teachings;

FIG. 17 b is a graphical schematic representation of the anotherestimator structure embodiment of these teachings applied to of a buckregulator;

FIG. 18 is a graphical representations of results obtained using theembodiment of FIG. 17 b;

FIG. 19 is a graphical schematic representation of yet another estimatorstructure embodiment of these teachings; and

FIG. 20 is a graphical representations of results obtained using theembodiment of FIG. 19.

DETAILED DESCRIPTION

In one embodiment, the system of these teachings includes a Switchingpower supply, an adaptive plant estimation component capable ofreceiving an output voltage from the Switching power supply and an inputto the Switching power supply and of providing a model of the Switchingpower supply; the model reflecting changes in output voltage state ofthe Switching power supply, a compensator design component capable ofreceiving the model of the Switching power supply and of providingcompensator parameters, the compensator parameters reflecting changes inoutput voltage state of the Switching power supply, an adaptivecompensator capable of receiving the compensator parameters and ofproviding the input control signal to the driver component. TheSwitching power supply includes a circuit having at least two reactivecomponents configured to provide an output voltage and capable of beingswitched from one output voltage state to another output voltage state,a switching component capable of switching said circuit between oneoutput voltage state and the other output voltage state, and a drivercomponent capable of receiving an input control signal and of diving theswitching component in order to cause switching between the one outputvoltage state and the other output voltage state in response to theinput control signal.

FIG. 2 presents a block diagram of an embodiment of the system of theseteachings. Referring to FIG. 2, a Switching power supply 110 includes aninductor 112 and a capacitor 114 and a switching component 116 and adriver component 118 (a PWM). The voltage across the capacitor 114 isthe output voltage of the Switching power supply 110. An adaptive plantestimation component 120 receives the output voltage 124 of theSwitching power supply 110 and the input control signal 126 to theSwitching power supply 110 and provides a model of the Switching powersupply 110, where the model reflects changes in the Switching powersupply 110. The model to Switching power supply 110 is provided to acompensator design component 130, which provides compensator componentparameters to the adaptive compensator 140. The compensator designparameters reflect changes in the compensation needed to account forchanges in the Switching power supply 110. In some embodiments, adisturbance injection component 150 provides a substantially small noisesignal that can aid in the detection of changes in the Switching powersupply 110. In some embodiments, the adaptive plant estimator component120 utilizes the LMS algorithm in order to provide a model of theSwitching power supply 110. (For a description of the LMS algorithm,see, for example, S. Haykin, Introduction to Adaptive Fillers, ISBN0-02-949460-5, pp. 108-110, which is incorporated by reference herein.).In other embodiments, the adaptive plant estimator component 120utilizes and RLS algorithm (for a description of the RLS algorithm, see,for example, S. Haykin, Introduction to Adaptive Filters, ISBN0-02-949460-5, pp. 139-143, which is incorporated by reference herein).

It should be noted that while the exemplary Switching power supplyembodiment of these teachings is described by one exemplary type, otherpower supply architectures such as boost, buck-boost, flyback, forward,etc are within the scope of these teachings.

In one embodiment, the controller of these teachings includes a samplingcomponent capable of sampling an output signal from a system and aninput signals from the system at a first sampling rate, the firstsampling rate being at least equal to a predetermined operating rate, aninput parameter obtaining component capable of receiving the outputsignal and the input signal sampled at the first sampling rate and ofobtaining values for a plurality of input parameters, the values for theinput parameters being sampled at the first sampling rate, a decimatorcomponent capable of receiving the values for the input parameterssampled at the first sampling rate and of providing subsampled valuesfor the input parameters, subsampled values being sampled at a secondsampling rate, the second sampling rate been slower than the firstsampling rate, an adaptive plant estimator component capable ofreceiving the subsampled values of the input parameters and of obtaininga model of the system, the model reflecting variations in the system.

Although the embodiments described hereinbelow are described in terms ofa particular controlled component, it should be noted that theembodiments described hereinbelow can be applied to a wide range ofother controlled components.

FIG. 3 shows a block diagram representation of an embodiment of thecontroller of these teachings. Referring to FIG. 3, the embodiment showntherein includes a sampling component 220 that samples an output signalfrom a system 210 and an input signal from the system at a firstsampling rate, the first sampling rate being greater than or equal to apredetermined operating rate, an input parameter obtaining component 230capable of receiving the output signal and the input signal sampled atthe first sampling rate and of obtaining values for a number of inputparameters, the values for the input parameters being sampled at thefirst sampling rate, a decimator component 240 capable of receiving thevalues for the input parameters sampled at the first sampling rate andof providing subsampled values for the input parameters, the subsampledvalues being sampled at a second sampling rate, the second sampling ratebeen slower than the first sampling rate, an adaptive plant estimatorcomponent 250 that receives the subsampled values of the inputparameters and obtains a model of the system 210, the model reflectingvariations in the system and a compensator design component 260 thatreceives the model of the system and of providing compensatorparameters, the compensator parameters reflecting changes in the system;values of said compensator parameters being sampled at the secondsampling rate; said compensator design component being capable ofproviding said values of said compensator parameter to a compensator270. The compensator 270 operates at the predetermined operating rate.

In one exemplary embodiment, these teachings not being limited to thatexemplary embodiment, parameters of the system 210 (DC-to-DC powersupply) vary slowly. Therefore it is possible to make the parameterupdates a slower, offline computation. In a linear compensator typedesign, an analog-to-digital converter 220 (ADC) measures the output andinput (and intermediate in some embodiments) voltages in the powersupply 210 and provides them to the compensator 270. This allows forboth error feedback and correction of input supply variations. The ADCresults are also used by the auto- and cross-correlators 230 to measurethe behavior of the power supply 210. The resulting correlationcoefficients are used to design the compensator. The parametercomputation and compensator design are done offline at a lower samplingrate. This lowers the cost of those tasks, because the digital logic canbe in the form of a simple micro-sequencer. If it is desired, thecompensator can also be implemented in analog form, with digitaladjustments made during the compensator design stage.

In another embodiment, shown in FIG. 4, the controller of theseteachings also includes a load current estimating component 280 capableof receiving the output signal sampled at the first sampling rate andstate estimation data from the adaptive plant estimator component 250and of providing estimated load current data at the first sampling rateand another decimator component 288 capable of receiving the estimatedload current data at the first sampling rate and of providing estimatedload current data at the predetermined operating rate to the compensator270.

In one instance, the ADC provides inputs to a state estimator 284. Theestimated states are then used by the feedback gain matrix in thecompensator 270 to complete the feedback system. In another instance, aload current estimator is also included. The load current estimator 286allows for the effect of variations in the load current to be minimized.The values from the load current 286 and the state estimator 284 areprovided to another decimator 288 that provides estimated load currentdata at the predetermined operating rate to the compensator 270.

In one instance, as shown in FIG. 5, the ADC is an oversampling ADC,aDelta Sigma ADC 290 in the embodiment shown, including a anoversampling modulator, delta-sigma modulator 294 in the embodimentshown, and a decimation filter 296. The oversampling modulator 290converts the analog input voltage into a high-speed, digital bit stream.The digital bit stream may be as little as one bit wide. Because thedigital bit stream is sampled at a very high rate, it is possible tolow-pass filter the bit stream and to recover a high-precision,lower-sample-rate representation of the analog signal.

In one embodiment, shown in FIG. 6, the sampling component (220, FIG. 3)is a oversampling (sigma delta in one embodiment) modulator 310 and thefirst sampling rate is an oversampled rate. In the embodiment shown inFIG. 6, the input parameter obtaining component (230, FIG. 3) is anautocorrelation and crosscorrelation estimator 320. It should be notedthat other embodiments of the input parameter obtaining component arepossible and within the scope of these teachings. It should also benoted that embodiments are possible in which the oversampling (sigmadelta in one embodiment) modulator 310 provides inputs to the stateestimator 284 and the load estimator 286.

In many applications, including the DC-to-DC converter application, andin particular for embodiment utilizing the cross- and autocorrelationfunctions, the decimation filter (decimator) function 240 can bebuilt-in. This reduces the cost because a one-bit multiplier is just asingle gate, while a high-precision digital multiplier can be a costlydesign.

FIG. 7 shows an embodiment of the system of these teachings in which theoversampling (sigma delta in one embodiment) modulator 310 is used as asampling component. In the embodiment shown in FIG. 7, the system is aswitching power supply or a generic power supply. The driver componentin the switching power supply is typically a PWM (pulse widthmodulator). Of specific interest in many instances, especially, but notlimited to, the instance in which the driver component is a PWM, it isthe specific embodiment in which the first sampling rate (or theoversampled rate in one instance) is substantially equal to twice theoperating rate of the system (the PWM rate in embodiments in which thedriver is a PWM). In the embodiment shown in FIG. 7, the decimator isintegral to the autocorrelation and crosscorrelation estimator 320.

FIG. 8 shows an embodiment of the system of these teachings in which theoversampling (sigma delta in one embodiment) modulator 310 is used as asampling component and load current and state estimation is alsoperformed. In the embodiment shown in FIG. 8. the decimator is integralto the autocorrelation and crosscorrelation estimator 320 and the otherdecimator (288, FIG. 4) is also built-in. The use of a high-speed bitstream for digital converter control applies not just to correlators butalso to compensators as well, in particular, even a simpleproportional-integral-derivative (PID) compensator, when implemented inthe velocity form. In this compensator, the error signal is processed bythe sum of a gain, a gain times the derivative, and a gain times thesecond derivative. The processing result is then integrated. Thedecimation function, in this embodiment, is achieved by the finalintegrator and the dynamics of the power supply itself.

In one embodiment, the method of these teachings includes sampling anoutput signal from a system and an input signal from the system at afirst sampling rate, the first sampling rate being at least equal to apredetermined operating rate, obtaining, from the sampled output signaland the sampled input signal, values for a number of input parameters,the values for the input parameters being sampled at the first samplingrate, decimating the values for the input parameter in order to obtainvalues for a number of subsampled input parameters, the subsampled inputparameters being sampled at a second sampling rate, the second samplingrate been slower than the first sampling rate, obtaining, from thesubsampled input parameters, a model for the system, obtaining, from themodel for the system, compensator parameters and providing thecompensator parameters to an adaptive compensator.

In one instance of the above described embodiment of the method of thepresent teachings, the input parameters for generating a model of thesystem are the autocorrelation and crosscorrelation. In one instance,simplified hardware can be used to calculate the auto- andcross-correlation. in many instances, the compensator does not have tobe updated at every cycle thus the calculations of the planned estimatecan be done at a much lower sampling rate. This allows the high-speedpart of the algorithm to be implemented in specialized hardware and thelow-speed part of the algorithm to be implemented in a very low-costgeneral purpose microprocessor.

In one instance, the high-speed algorithm may use a delta-sigmamodulator as the ADC conversion element, as disclosed hereinabove. Thishigh-speed, small-bit-width, over-sampled data conversion method allowsfor simpler hardware. The typical delta-sigma ADC decimator can beintegrated into or replaced by the correlation filter. Thus thecorrelation filter hardware is simplified.

A number of conventional techniques (algorithms) can be utilized for theadaptive identification of unknown dynamic systems. One of theseconventional techniques is the least mean squares (LMS) algorithm. Thistechnique can be easily implemented. However, the LMS algorithm can beslow to converge. The LMS algorithm is effective for the power supplyconverter application because good initial guesses can be given, so thepower supply behaves properly before any adaptation is achieved. In manyapplications, it is desirable to use an algorithm that can identify thedynamics of the power supply in a time short compared to the time overwhich the system changes. For example, it is desirable for the algorithmto determine the dynamics of the object being controlled, in this case,the power electronics associated with the power supply, during startupand before regulation began.

One conventional technique for fast plant identification is theRecursive Least Squares (RLS) algorithm. This technique provides forfast convergence, however, it can suffer from a high computationalburden. Also, the normal formulation of the Recursive Least Squaresalgorithm also has some assumptions in it that limit very high-speedperformance.

The foundation of the use of autocorrelation and crosscorrelation forsystem identification comes from the statistical solution whichminimizes the mean square error of the Wiener filter is given inequation (1) where ŵ is the vector of estimated filter coefficients,R_(xx) is the correlation matrix of the input signal, and r_(xy) is thecross-correlation vector of the input and output signals.

ŵ=R _(xx) ⁻¹ r _(xy)   (1)

It is possible to numerically estimate the auto-correlation matrix andcross-correlation vector from the observed input and output signals ofthe plant (system) being identified. These estimates may be useddirectly to compute the estimated weight vector. In one embodiment ofthe method of the present teachings, the computational load is separatedinto two segments, one which must be performed on every new data sample,and one which must be performed only when it is desired to update theweight estimates. By scheduling the weight updates at a slower rate thanthe data, the overall computational burden of the algorithm can bereduced. However, when computed, the weight estimates will still makeuse of all data to that point. Thus, this method sacrifices only theweight update rate and not the quality of the estimates. Equation (2)depicts numerical estimates of the auto-correlation matrix andcross-correlation vector. The individual terms can then be computedusing equations (3) and (4). Additionally, these expressions can berewritten in a recursive manner such that they are updated incrementallywith each new data point as given by (5) and (6), where the second indexis the discrete time offset.

$\begin{matrix}{{{\hat{R}}_{xx} = \begin{bmatrix}{{\hat{r}}_{xx}\lbrack 0\rbrack} & {{\hat{r}}_{xx}\lbrack 1\rbrack} & \ldots & {{\hat{r}}_{xx}\left\lbrack {P - 1} \right\rbrack} \\{{\hat{r}}_{xx}\lbrack 1\rbrack} & {{\hat{r}}_{xx}\lbrack 0\rbrack} & \; & {{\hat{r}}_{xx}\left\lbrack {P - 2} \right\rbrack} \\\vdots & \; & \ddots & \; \\{{\hat{r}}_{xx}\left\lbrack {P - 1} \right\rbrack} & {{\hat{r}}_{xx}\left\lbrack {P - 2} \right\rbrack} & \; & {{\hat{r}}_{xx}\lbrack 0\rbrack}\end{bmatrix}},{{\hat{r}}_{xy} = \begin{bmatrix}{{\hat{r}}_{xy}\lbrack 0\rbrack} \\{{\hat{r}}_{xy}\lbrack 1\rbrack} \\\vdots \\{{\hat{r}}_{xy}\left\lbrack {P - 1} \right\rbrack}\end{bmatrix}}} & (2) \\{{{\hat{r}}_{xy}\lbrack j\rbrack} = {\frac{1}{N}{\sum\limits_{m = 0}^{N - 1}{{y\lbrack m\rbrack}{x\left\lbrack {m - j} \right\rbrack}}}}} & (3) \\{{{\hat{r}}_{xx}\lbrack j\rbrack} = {\frac{1}{N}{\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - j} \right\rbrack}}}}} & (4) \\{{{\hat{r}}_{xy}\left\lbrack {j,{k + 1}} \right\rbrack} = {{\frac{k + 1}{k + 2}{{\hat{r}}_{xy}\left\lbrack {j,k} \right\rbrack}} + {\frac{1}{k + 1}{y\lbrack k\rbrack}{x\left\lbrack {k - j} \right\rbrack}}}} & (5) \\{{{\hat{r}}_{xx}\left\lbrack {j,{k + 1}} \right\rbrack} = {{\frac{k + 1}{k + 2}{{\hat{r}}_{xx}\left\lbrack {j,k} \right\rbrack}} + {\frac{1}{k + 1}{x\lbrack k\rbrack}{x\left\lbrack {k - j} \right\rbrack}}}} & (6)\end{matrix}$

The optimal weight vector based on the numerical estimates can then beexpressed as given in equation (7). Since the auto-correlation matrixhas a Toeplitz structure, there are only 2P recursive estimatesnecessary for computation of the entire expression. Additionally, thisstructure allows for the use of efficient matrix inversion techniqueswhich utilize this symmetry to reduce the number of requiredcomputations.

$\begin{matrix}{\hat{w} = {\begin{bmatrix}{\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\lbrack m\rbrack}}} & {\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} & {\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}} \\{\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} & {\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\lbrack m\rbrack}}} & {\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} \\{\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}} & {\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} & {\sum\limits_{m = 0}^{N - 1}{{x\lbrack m\rbrack}{x\lbrack m\rbrack}}}\end{bmatrix}^{- 1}\mspace{529mu}\begin{bmatrix}{\sum\limits_{m = 0}^{N - 1}{{y\lbrack m\rbrack}{x\lbrack m\rbrack}}} \\{\sum\limits_{m = 0}^{N - 1}{{y\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} \\{\sum\limits_{m = 0}^{N - 1}{{y\lbrack m\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}}\end{bmatrix}}} & (7)\end{matrix}$

In another embodiment, the method of these teachings includes samplingan output signal from a system and an input signal from the system,obtaining, from the sampled output signal and the sampled input signal,values for a predetermined finite number of rows and columns from aninverse matrix and a predetermined finite number for a row vector in aleast-squares solution, and obtaining, from the values for thepredetermined finite number of rows and columns from an inverse matrixand the predetermined finite number for a row vector in a least-squaressolution, a model for the system. Once a model of the system isobtained, an adaptive control method can be implemented.

While not desiring to be bound by theory, one rationale motivating theabove described embodiment is given hereinbelow. The result of equation(7) above provides the same answer as the batch least squares solutionin the case of infinite data. However, for any finite interval of data,the above result and the batch least squares result will differ. Thiscan be seen directly by computing the batch mode least squares solutiongiven by (8) over the same time interval [0,N−1], which results in theleast squares weight vector given in equation (9). Comparison of thesolutions in equation (7) and (9) indicate differences in theauto-correlation matrices. In fact, the least squares auto-correlationmatrix of equation (9) is not necessarily Toeplitz for any finite dataset. For a stationary input signal, and as the numerical sum approachesthe true statistical value, that the two solutions (Equations (7) and(9)) are identical. However, in many applications, it is desirable tohave the best possible estimate given a short finite interval of data.The estimate given by (9) minimizes the sum squared error over anyfinite data set.

$\begin{matrix}{\mspace{79mu} {\hat{w} = {\left\lbrack {X^{T}X} \right\rbrack^{- 1}X^{T}y}}} & (8) \\{\hat{w} = {\begin{bmatrix}{\sum\limits_{m = 2}^{N - 1}{x\lbrack m\rbrack}^{2}} & {\sum\limits_{m = 2}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} & {\sum\limits_{m = 2}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}} \\{\sum\limits_{m = 2}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} & {\sum\limits_{m = 2}^{N - 1}{x\left\lbrack {m - 1} \right\rbrack}^{2}} & {\sum\limits_{m = 2}^{N - 1}{{x\left\lbrack {m - 1} \right\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}} \\{\sum\limits_{m = 2}^{N - 1}{{x\lbrack m\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}} & {\sum\limits_{m = 2}^{N - 1}{{x\left\lbrack {m - 1} \right\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}} & {\sum\limits_{m = 2}^{N - 1}{x\left\lbrack {m - 2} \right\rbrack}^{2}}\end{bmatrix}^{- 1}\mspace{529mu}\begin{bmatrix}{\sum\limits_{m = 2}^{N - 1}{{y\lbrack m\rbrack}{x\lbrack m\rbrack}}} \\{\sum\limits_{m = 2}^{N - 1}{{y\lbrack m\rbrack}{x\left\lbrack {m - 1} \right\rbrack}}} \\{\sum\limits_{m = 2}^{N - 1}{{y\lbrack m\rbrack}{x\left\lbrack {m - 2} \right\rbrack}}}\end{bmatrix}}} & (9)\end{matrix}$

The individual terms in equation (9) can, in one embodiment, berepresented by iterative solutions and the computational load can besplit as discussed above such that the weight vector solution isperformed at a sub-sampled rate. However, there are now

$\frac{1}{2}\left( {P^{2} + P} \right)$

unique entries in the auto-correlation matrix. It is not necessary tocompute each of these separately since elements along each diagonal aresimply a delayed version of first element of the diagonal as given byequation (10).

a _(i,l) [k]=a _(i−1,j−1) [k−1]  (10)

Given the above described property, and the symmetry of the matrix, onlyP running estimates need to be computed and the appropriate past valuesof each estimate must be stored. The auto-correlation matrix can then becomputed directly from these values. Hereinbelow, this embodiment of themethod of the present teachings is also referred as Iterative LeastSquares since it exactly implements the least square solution over anyfinite interval of data.

The above described embodiment is generally applicable for both the FIRand IIR filters. In the instance where the above described embodiment isapplied to IIR filters, the input matrix includes a mixture of both thepast and current inputs as well as the past outputs.

It should be noted that the embodiments described hereinabove can beapplied to a wide range of controlled components besides switching powersupplies.

In order to better illustrate the systems and methods of the presentteachings, results and details of several exemplary embodiments arepresented hereinbelow. It should be noted that the methods and systemsof these teachings are not limited to (or limited by) the illustrativeembodiment presented hereinbelow.

In one illustrative embodiment, the Iterative Least Squares solution ofequation (9) is compared with the method of equation (7) which uses theconventional numerical correlation estimates. In one instance, the twoembodiments are simulated in a noise-free environment. The system to beidentified is a 5 tap FIR filter with coefficients given in (9) and fedwith a uniformly distributed random number sequence.

B=[1.0 2.0 −0.5 −0.3 1.2]  (11)

FIG. 9 depicts, for the instance of a noiseless environment, two of thefilter coefficients versus time. The dashed horizontal lines are for theIterative Least Squares solution while the solid jagged lines for theconventional numerical correlation estimate method. In the noiselesscase simulated, the Iterative Least Squares solution is correct after 5time steps (the FIR filter length) while the solution based on thenumerically calculated correlations takes more iterations to converge.FIG. 11 depicts the sum squared error between the actual and estimatedweight vector at each time step for the noiseless case. In FIG. 11, thesolid line represents the conventional correlation estimates and thedashed or lower line is for the Iterative Least Squares solution.

In one instance, the Iterative Least Squares solution conventionalnumerical correlation estimate solution and are simulated for the casewhere the observations (filter outputs) are corrupted byuniformly-distributed random noise with an amplitude of 10% of theprimary excitation signal. Results of the simulation for two of thefilter coefficients versus time are shown in FIG. 10. FIG. 12 depictsthe above described error signals for the instance where 10% whitemeasurement noise is present.

In another illustrative embodiment, a controller was designed byassuming a second-order compensator and solving for the closed-looptransfer function. The denominator of the closed-loop transfer functionwas then equated to the desired transfer function. (The polynomialequation obtained by equating the desired transfer function to thedenominator of the closed-loop transfer function is known as theDiophantinc equation. See for example, Kelly, A., Rinne, K, Control ofdc-dc converters by direct pole placement and adaptive feedforward gainadjustment, Twentieth Annual IEEE Applied Power Electronics Conferenceand Exposition, 2005; APEC 2005, Volume 3, Date: 6-10 Mar. 2005, Pages:1970-1975, which is incorporated by reference herein.)

FIG. 13 depicts the transient response of a deadbeat controller,designed utilizing the methods of these teachings, to a step change inload current from 10 A to 13 A, In FIG. 13, the upper trace is outputvoltage and the lower trace is the commanded duty cycle. FIG. 14 depictsthe response of the deadbeat controller, designed utilizing the methodsof these teachings, to a large load current step from 5 A to 25 A wherethe control signal saturates at full duty cycle. (In FIG. 14, the uppercurve reflects output voltage, and the lower curve reflects commandedduty cycle).

As described herein above, the state estimator provides a model of thesystem to be controlled with an extra input that is used to drive theerror between the measurable variables and their correspondingquantities in the model. If the states are observable, it is possible todesign a gain for this extra input that results in a stable estimator.However, zero error between the estimated states and the real onmeasurable states is not always obtained, particularly in the presenceof unknown disturbances. This is a limitation when applying state spaceestimation in general and in particular when applying state spaceestimation for power supplies. Several embodiments of the stateestimator used in these teachings are disclosed hereinbelow.

Switching power supplies can be modeled by a set of differentialequations with time as the independent variable. As an example, thedifferential equations used to model a buck power supply are as follows:

$\frac{x}{t} = {{A_{e}x} + {B_{e}u}}$ y = C_(e)x + D_(e)u

Where: u is the input vector

-   -   x is the state vector    -   y is the output vector    -   A_(e), B_(e), C_(e), D_(e) are gain matrices

Assuming that PWM (while in one stationary mode) is modeled as aconstant gain equal to the supply voltage, the state vector is thecapacitor voltage and conductor current, the input is the duty cyclecommand, and the output is the output voltage, than the gain matricesare as follows:

${\frac{}{t}\begin{bmatrix}v \\i\end{bmatrix}} = {{{\begin{bmatrix}0 & \frac{1}{C} \\\frac{- 1}{L} & {- \frac{{Rc} + {Rl}}{L}}\end{bmatrix}\begin{bmatrix}v \\i\end{bmatrix}} + {{\begin{bmatrix}0 & \frac{Vdd}{L}\end{bmatrix}\lbrack{duty}\rbrack}\lbrack{vout}\rbrack}} = {{\begin{bmatrix}1 & {Rc}\end{bmatrix}\begin{bmatrix}v \\i\end{bmatrix}} + {\lbrack 0\rbrack \lbrack{duty}\rbrack}}}$

Where: v is the capacitor voltage

-   -   i is the inductor current    -   Vdd is the supply voltage    -   duty is the PWM duty cycle    -   L is the inductor inductance    -   C is the capacitor capacitance    -   Rc is the parasitic resistance of the capacitor    -   Rl is the parasitic resistance of the inductor.

The Ae, Be, Ce, and De gain matrices can be readily identified from theequations above. The additional input for state estimation is the errorbetween them measured values and their modeled values this error ismultiplied by the gain L, and added to the state vector. In thisembodiment, the first equation is modified as:

$\frac{x_{est}}{t} = {{A_{e}x_{est}} + {B_{e}u} + {L_{e}\left( {y - y_{est}} \right)}}$y_(est) = C_(e)x_(est) + D_(e)u

Where: x_(est) is the estimated states

-   -   yest is the estimate output    -   Le is the estimator gain matrix

Le can be calculator by conventional techniques.

FIG. 15 is a graphical schematic presentation of a state spaceestimator. The results of a exemplary simulation using the buckconverter equations can be seen in FIG. 16. It can be seen that there isa DC error in the inductor current. Also, the load current is notdetermined.

Another embodiment of the estimator of these teachings uses a differentstructure for the state space estimator where instead of using dutycycle as input, it attributes the error between the modeled outputvoltage and the measured output voltage primarily to an unknown loadcurrent. Using load current instead of the duty cycle input as input,the DC error is significantly reduced. Also, proportional integralcontrol is used to drive the average DC error between the output voltageand the modeled output voltage. FIG. 17 a is a graphical schematicrepresentation of another estimator structure embodiment of theseteachings,

Referring to FIG. 17 a, there are two inputs duty cycle 410 and outputvoltage 420. The PWM or pulse with modulator is modeled as a constantgain 430 and its output is driven into a model of the switching powersupply 460. The difference between the estimated output voltage and themeasured output voltage, the output of substractor 440, is used to drivea compensator 450, which in the embodiment shown, is a proportionalintegral compensator. The compensator output is an estimate 480 of theload current disturbance and is fed into the load current input of theswitching power supply model 460. The feedback action insures that therequired estimated load disturbance is generated by the compensator. Theswitching power supply model 460 also provides an estimate of the state470 and an estimate of the output voltage 490.

FIG. 17 b shows the discrete time model of a typical buck converter, oneinstance of the switching power supply model 460. The first integrator510 is used to model the inductor current and the second integrator 520is used to model the capacitor voltage. The inductor current and loadcurrent are summed to generate the capacitor current. Note that the signof the inductor current is chosen to match that of a typical load. Thatmeans an increase in inductor current causes a reduction in the outputvoltage. The loops containing Rind and Rcap model the losses and outputvoltage effects of the parasitic resistances.

FIG. 18 is a graphical representations of results obtained, in anexemplary instance, using the embodiment of FIGS. 17 a, 17 b. Note thatthe output voltage is nearly accurately estimated, the inductor currentsubstantially contains the expected DC information and the load currentis estimated.

FIG. 19 shows an additional embodiment, in which the duty cycle is takenas arising from a digital compensator and the discrete time convertermodel, low-pass filter and estimator compensator are implementeddigitally. In this embodiment, the digital sampling rate is chosen to bea multiple of the PWM rate. The combination of that comparator andlow-resolution DAC provides a noisy over sampled version of theestimation error. Analog components are the comparator and DAC. As analternative, a sigma delta modulator or other over-sampling modulatorcould be used. In this approach, the low-pass filter is the plant modeland no additional large filtering delays are incurred. FIG. 19 is anexemplary embodiment of an over-sampling state estimators that use sigmadelta or other over-sampling modulator, such as the comparator and DAC,in the error feedback path. Such an embodiment has application tocontrol systems other than control of switching power supplies and canresult in a smaller delay than other embodiments.

FIG. 20 is a graphical representations of results obtained, for anexemplary instance, using the embodiment of FIG. 19.

It should be noted that, although the above description of the teachingsutilized buck converters as an exemplary embodiment, these teachings arenot limited to that embodiment. Also, although the above description ofthe teachings utilized sigma delta modulator as an exemplary embodimentof over-sampling modulators, other over-sampling modulators are withinthe scope of this invention. (See, for example, but not limited to, theover-sampling modulators disclosed in U.S. Application Publication No.2______, corresponding to U.S. patent application Ser. No. 11/550,893,entitled Systems and Methods for Digital Control, both of which areincorporated by reference herein.)

Although the present teachings have been described with respect tovarious embodiments, it should be realized these teachings are alsocapable of a wide variety of further and other embodiments within thespirit of the present teachings.

What is claimed is:
 1. A system comprising: a power supply modelcomponent that receives an estimated disturbance; and a disturbanceestimating component that provides said estimated disturbance from adifference between a sensed output of a power supply and an estimatedoutput from said power supply model component.
 2. The system of claim 1wherein: said disturbance estimating component comprises a proportionalintegral component that provides said estimated disturbance, where saidestimated disturbance includes an estimated load current; and said powersupply model component receives a duty cycle multiplied by apredetermined gain and provides an estimated state of said power supplyand said estimated output based on said duty cycle.
 3. The system ofclaim 1 further comprising an oversampling modulator that provides saiddifference between said sensed output and said estimated output, whereinsaid difference includes an oversampled difference provided to saiddisturbance estimating component, and wherein said disturbanceestimating component outputs an oversampled estimated disturbance. 4.The system of claim 3 further comprising a low pass filter that filterssaid oversampled estimated disturbance and that provides said estimateddisturbance.
 5. A system comprising: a component that provides adifference between a sensed output and an estimated output of saidsystem; and an oversampling modulator that provides an oversampleddifference between said sensed output and said estimated output, whereinsaid component provides an estimated state of said system and saidestimated output based on said oversampled difference.
 6. A methodcomprising: obtaining a difference between a sensed output of a powersupply and an estimated output from a power supply model; obtaining fromsaid difference an estimated power supply disturbance; and providingsaid estimated power supply disturbance to said power supply model. 7.The method of claim 6, wherein said estimated power supply disturbanceincludes an estimated load current, the method further comprising:providing a duty cycle multiplied by a predetermined gain to said powersupply model; and providing an estimated state of said power supply andsaid estimated output from said power supply model based on said dutycycle.
 8. The method of claim 6 further comprising oversampling saiddifference and providing an oversampled estimated disturbance.
 9. Themethod of claim 8 further comprising filtering said oversampledestimated disturbance and providing said estimated power supplydisturbance.
 10. A method comprising: obtaining a difference between asensed output from a system and an estimated output from a system model;oversampling said difference; and reducing an error in said estimatedoutput based on said oversampled difference.